📚 Mastering Vectors at KS3 | KS3 向量考点精讲
Vectors are a key topic in KS3 mathematics, introducing the idea of quantities that have both magnitude and direction. Unlike scalars, which only need a number and a unit, vectors describe movements, forces, and positions in a more complete way. This revision guide covers all essential concepts you need to master vectors at Key Stage 3.
向量是 KS3 数学中的重要主题，引入既有大小又有方向的量。与只需数值和单位的标量不同，向量能更全面地描述运动、力和位置。本复习指南涵盖你在关键阶段 3 需要掌握的所有向量核心概念。
1. What is a Vector? | 什么是向量？
A vector is a quantity that has both magnitude (size) and direction. For example, velocity and displacement are vectors, while speed and distance are scalars. Understanding this difference is fundamental to working with movement and forces.
向量是既有大小（量值）又有方向的量。例如，速度和位移是向量，而速率和距离是标量。理解这一差异是使用运动与力的基础。
We usually represent a vector by an arrow; its length shows the magnitude, and the arrowhead shows the direction. A longer arrow means a larger quantity, and the arrow points exactly where the vector acts or moves.
我们通常用箭头表示向量；箭头的长度表示大小，箭头指向表示方向。较长的箭头意味着数量较大，箭头精确指向向量作用或移动的方向。
Common vectors in KS3 include displacement (‘3 km east’), velocity (‘5 m/s north’), and force (’10 N downwards’). Any quantity that needs both ‘how much’ and ‘which way’ is a vector.
KS3 常见的向量包括位移（’向东 3 千米’）、速度（’向北 5 米/秒’）和力（’向下 10 牛’）。任何需要同时说明’多少’和’哪个方向’的量都是向量。
2. Representing Vectors | 向量的表示方法
In print, vectors are often written in bold, such as a; in handwriting, we put a tilde or an arrow above, like →a or a . Recognising these notations helps you read exam questions correctly.
在印刷体中，向量常用粗体表示，如 a；在手写时，我们在字母上方加波浪线或箭头，如 →a 或 a 。识别这些记号有助于正确理解考题。
A vector from point A to point B is written as AB with an arrow above, or simply AB in bold. This tells you both the starting point and the end point of the displacement.
从点 A 到点 B 的向量可以标注为 AB 上加箭头，或粗体 AB。这同时告诉你位移的起点和终点。
A very useful numerical form is the column vector, which lists the horizontal component above the vertical component. For example, a vector that moves 3 units right and 4 units up can be written as the column vector
| 3 |
| 4 |
.
一种非常有用的数值形式是列向量，它将水平分量写在垂直分量上方。例如，向右移动 3 单位、向上移动 4 单位的向量可写成列向量
| 3 |
| 4 |
。
3. Column Vectors | 列向量
A column vector
| x |
| y |
describes movement: the top number x tells you how far to go horizontally (positive for right, negative for left), and the bottom number y tells you how far to go vertically (positive for up, negative for down).
列向量
| x |
| y |
描述移动：顶部数字 x 表示水平方向移动距离（正数向右，负数向左），底部数字 y 表示垂直方向移动距离（正数向上，负数向下）。
For instance,
| 2 |
| 5 |
means 2 units right and 5 units up, while
| -3 |
| 1 |
means 3 units left and 1 unit up. You can use these components directly in calculations.
举例来说，
| 2 |
| 5 |
表示向右 2、向上 5，而
| -3 |
| 1 |
表示向左 3、向上 1。你可以在计算中直接使用这些分量。
When both components are zero, you have the zero vector
| 0 |
| 0 |
, which has no direction and does not cause any movement.
当两个分量都为零时，你得到零向量
| 0 |
| 0 |
，它没有方向，也不会产生任何移动。
4. Magnitude of a Vector | 向量的大小
The magnitude (length) of a vector v =
| x |
| y |
is found using Pythagoras’ theorem: |v| = √(x² + y²). This gives the straight-line distance from the start to the end of the arrow.
向量 v =
| x |
| y |
的大小（模长）使用勾股定理计算：|v| = √(x² + y²)。这给出了从箭头起点到终点的直线距离。
For vector a =
| 3 |
| 4 |
, its magnitude is √(3² + 4²) = √25 = 5. If the components are in metres, the magnitude is 5 metres.
对于向量 a =
| 3 |
| 4 |
，其大小为 √(3² + 4²) = √25 = 5。若分量的单位是米，则大小就是 5 米。
Magnitude is always a non-negative number. Even if a vector has negative components, squaring them makes the result positive, so you never get a negative length.
大小永远是一个非负数。即使向量的分量有负数，平方后结果为正，因此你不会得到负的长度。
5. Adding Vectors | 向量加法
To add two vectors a and b, you can use the triangle law: place the tail of b at the head of a; the sum a + b is the vector from the tail of a to the head of b. This works for any two vectors.
要相加两个向量 a 和 b，可以使用三角形法则：将 b 的尾端置于 a 的首端；和向量 a + b 是从 a 的尾端指向 b 的首端的向量。这适用于任意两个向量。
Alternatively, use the parallelogram law: draw both vectors from the same starting point, complete a parallelogram, and the diagonal from the common point gives a + b.
也可以使用平行四边形法则：从同一起点画出两个向量，补全一个平行四边形，从共同起点画出的对角线即为 a + b。
With column vectors, addition is simply done by adding corresponding components:
| x₁ |
| y₁ |
+
| x₂ |
| y₂ |
=
| x₁+x₂ |
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